Tom Rocks Maths is back on Oxide Radio for Hilary Term 2019 with the usual eclectic mix of maths and music. Learn more about the only million-dollar Millennium Problem that’s been solved so far, fun facts about the number 6, and a nursery rhyme themed puzzle. Plus, music from Bring me the Horizon, Queen and Papa Roach. This is maths, but not as you know it…
Sir Michael Atiyah Riemann Hypothesis Proof
Sir Michael Atiyah explains his proof of the infamous Riemann Hypothesis in one slide. Recorded live at the Heidelberg Laureate Forum 2018.
Navier-Stokes: one equation to rule them all
A leading Millennium Prize Problem is the Navier-Stokes equation, which, if solved, could model the flow of any fluid – that means how aeroplanes navigate the skies, how water meanders in a river and how the flow of blood courses through your blood vessels… Understanding these equations in more detail will lead to scientific advances in all of these fields: better aircraft design, improved flood defences, and better drug delivery in the body. Fluids expert and mathematician Keith Moffatt took me down to the deep dark depths of Cambridge’s maths lab…
- For most fluids, including air and water, the Navier-Stokes equations are based on Newton’s Laws and were first written down in the 19th century
- The millennium problem is to answer the question of whether or not the equations can become infinite
- It cannot be solved with a computer because a computer programme will break down before the singularity at infinity is reached
- A real-world example is when two tornado-like vortices collide and undergo a process called ‘vortex reconnection’
You can listen to the full interview for the Naked Scientists here.
And you can read more about the Navier-Stokes equations and all of the Millennium Problems here.
Perelman and the Poincaré Conjecture
Grigori Perelman is a quiet and unassuming mathematician from Russia, who took the world of maths by storm in 2010 when he not only solved the Poincare problem but then refused the $1 million reward! I went along to the Millennium Bridge in London to meet mathematician Katie Steckles to shed some light on Perelman’s story and to find out why the Millennium Bridge was in fact its own millennium maths problem…
- When the Millennium Bridge opened its resonant frequency matched that of walking pedestrians which caused it to vibrate massively as seen in the video below
- In the field of topology things are considered equal if you can get from one to the other by doing a ‘smooth and gradual change’
- The Poincare Conjecture states that any shape satisfying a set of three conditions can be deformed into a sphere, and this will hold true in any number of dimensions
- It had been proved for all dimensions except 4, which was shown to be true by Grigori Perelman in 2002
- He published his proof on the internet and then refused the $1 million prize money, instantly becoming a sensation
You can listen to the full interview for the Naked Scientists here.
Solve this and you’ll win $1 million
My first ever live radio interview from July 2015 – enjoy! You can listen to the full interview for the Naked Scientists here.
The Millennium Prize Problems are a set of 7 maths problems that have been deemed so important that if you can solve any of them, you’ll be awarded 1 million Dollars. I was interviewed by Naked Scientist presenter Graihagh Jackson to explain exactly what the problems entail…
- The Millennium problems are a second reincarnation of the idea of important maths problems with the first set of 23 being proposed by Hilbert in 1900.
- The prizes are offered by the Clay Institute and so far only one of the seven has been correctly solved
- The 7 problems are the Navier-Stokes Equations, the Mass Gap Hypothesis, the Poincare Conjecture, the Riemann Hypothesis, P vs NP, the Birch and Swinnerton-Dyer Conjecture and the Hodge Conjecture
- Estimates of the time required to solve one of the problems actually results in being paid below minimum wage
If you want to find out more about the Millennium Problems you can find a series of articles I’ve written on the subject here.
The Poincare Conjecture
I’ve saved the best until last because this one’s been solved! Hallelujah! Praise the Lord! God save the Queen! Slap my thighs and serve me a milkshake! And the story of the man that did it is fascinating. But we’ll get to that… first up I’d better tell you what the problem is/was.
For the Poincare Conjecture we venture into the shape-shifting world of topology. This is a real favourite amongst mathematicians because it’s great for blowing people’s minds. The classic example: in topology a donut and a teacup are the same item. Yes, you did read that correctly. The reason? They both only have one hole: in the centre of the donut and in the centre of the handle of the teacup. It’s the number of holes that’s key. If you have a muffin and a donut how would you tell them apart? And no you can’t eat them. In topology what you would do is take an elastic band and put it around each object. Then you squeeze it in tightly until the object becomes one ball of mass. Well the muffin does, but the donut doesn’t. Not without breaking it at least – that’s the key. The hole in the donut means that you can’t shrink it and squeeze it down into one little ball without breaking it somehow. There’s no way to remove the hole.
If you’re still struggling to get to grips with topology, think of it like this: you have a donut made out of Playdoh and you need to mould it into a teacup with the only rule being that you cannot create or destroy any holes. You can do it. It’s a little fiddly yes, but it can be done. Now imagine you need to make a muffin with the same rule. No hole destroying. You can’t do it. You’ll always be left with a loop of Playdoh hanging off your perfectly crafted muffin.
The Poincare conjecture is based on this same idea: imagine you have a smooth shape made out of Playdoh and it has no holes, then the question is: can we mould it to make a sphere? Sounds easy enough in our three dimensional world, you can literally get a ball of Playdoh and make any shape you want – you can always get back to a sphere. But what happens in higher dimensions? Time is often referred to as the fourth dimension in Physics, but what comes after that? We as humans are not programmed to visualise it, but in maths higher dimensions exist. The interesting thing with this problem is that we know you can make a sphere in any number of dimensions except four and this is what the Poincare Conjecture asks. Can you take any four-dimensional smooth object that doesn’t contain any holes and turn it into a sphere? Turns out you can, just ask Gregori Perelman (photo credit: George M. Bergman – Mathematisches Institut Oberwolfach (MFO), GFDL, https://commons.wikimedia.org/w/index.php?curid=11511619).
Perelman is a Russian mathematician and he is quite the character. He showed that the Poincare Conjecture was true and then without really telling anyone just posted his solution online in 2002. No big event, no announcement, just a casual ‘oh here’s what I’ve been working on the past few years – I solved a Millennium Problem’. You have to love him for it. And it gets better. It took the Clay Institute eight years to verify that his solution was indeed correct and Perelman did not like this, not one bit. He couldn’t understand why they had to check his work – he is a mathematician; he doesn’t make mistakes! When the time came around for him to be presented with his money he declined, flat out turned it down. He was also awarded the maths version of the Nobel Prize, the Field’s Medal, and didn’t want that either. He was so annoyed at the way in which he was treated following his work that he gave up the subject and it is rumoured that he now works in Computer Science. Get your bets in now that he solves the biggest problem in that subject within the next few years…
This is a nice story for me to end on as it brings me back full circle to my starting point for these articles. The Millennium Maths problems were the first set of problems to get me really excited about maths. Whether it was because of the money or just the idea that these things even existed, I don’t know (it was probably the money), but what I do know is that Gregori Perelman is the perfect example of everything that is great about mathematicians. He started working on the Poincare Conjecture in 1995 before it was even a Millennium Problem, and then he turned down all of the fortune and fame that came with his solution. He simply wanted to be left alone to ‘do the maths’.
If after reading my articles you were thinking of attempting to solve one of these problems yourself by all means get stuck in, but as I started with a word of warning about the difficulty of these problems, let me end with another. Estimates of the number of hours spent by Perelman in solving the Poincare Conjecture actually put the $1 million prize money at less than the minimum wage. You have to love the subject to tackle these problems and I hope that I have and will continue to help you do exactly that.
You can listen to me talking to mathematician Katie Steckles about the Poincare Conjecture here.
I’ve written a series of articles on each of the 7 Millennium Problems which can be found here.
The Hodge Conjecture
The sixth Millennium Problem pretty much sums up why a lot of people don’t like maths. It is by far the hardest to explain in any terms, never mind simple ones, it is incredibly far out of reality and everyday experiences and mathematicians can’t agree on what the actual problem is – never mind how to go about trying to find a solution. I just want to emphasise this last point: depending on which mathematician you ask to define the problem; you will most likely get a different version of what it actually is. The official statement on the Clay Institute website sums it up perfectly: “The answer to this conjecture determines how much of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations.” So if we have a solution to a set of equations, is that solution also the solution to another different set of equations. Solution, solution, solution.
Not to be deterred I will try my best to explain the Hodge conjecture to you – just maybe lower your expectations a little. Instead of that 150-year aged Scotch whiskey you’re used to, on this occasion you might have to settle for a blue WKD (Jamie Vardy would be pleased at least). Before we get to the problem let’s start with the man that proposed it: William Hodge. Hodge was born in Scotland in 1903, he studied at Edinburgh, then Cambridge where he was appointed to a chair position which he held until his death in 1970. He won some awards, including being knighted by the Queen in 1959, and his conjecture was announced in 1950. That’s pretty much it, the full life story. It would be fair to say that we know as much about the man as we do about his conjecture…
I’ve probably stalled long enough, now onto the conjecture itself. In as simple terms as possible, the Hodge conjecture asks whether complicated mathematical things can be built from simpler ones. Not so dissimilar to seeing an entire working city built from Lego and realising that it is in fact all just made from little simple square blocks. Hodge is basically asking whether maths is the same as Lego. The Lego Movie brought the toy back to life so maybe Hollywood should do a Maths Movie to bring the subject into the 21st century. Mila Kunis and Brad Pitt in a race against time to work out the area of a circle before the world blows up… I’d watch it.
To explain the Hodge conjecture in a little more detail I need to give you another quick history lesson. In the first half of the 1900’s mathematicians began to realise that highly complicated mathematical objects could be approximated very well by sticking together smaller and more simple building blocks. The idea being let’s take something we don’t understand and try and build it using something we do. You can imagine the following scenario: an archaeologist realises that his evolution tree is missing a dinosaur – there’s no way the T-rex evolved into a chicken, there must be something in-between. The archaeologist has no idea what this dinosaur looks like, but can start using the information he has about other dinosaurs, for example other fossils and skeletons that were from the same time period to build up a picture of what the missing dinosaur might have looked like…
An artist’s impression:
It’s only a guess and almost certainly isn’t Big Bird, but after more searching and digging for fossils they might find something pretty similar to that guess. This is what mathematicians do. We use what we know to get a better understanding of the things that we don’t know much about.
Now with this idea in mind, we can finally come back to the Hodge conjecture. It asks whether one important class of mathematical objects (called projective algebraic varieties if you must know), which are made up of pieces called Hodge cycles, are in fact themselves made up of smaller pieces called algebraic cycles. In the Lego scenario, the problem is asking whether the Lego blocks making up the city are in fact made up of even smaller and simpler pieces. I did warn you this one gets a little tricky but hopefully now that we’ve got to the end “everything is awesome!!!”
I’ve written a series of articles on each of the 7 Millennium Problems which can be found here.
The Birch and Swinnerton-Dyer Conjecture
Next up is the Birch and Swinnerton-Dyer conjecture – somehow seems an apt name though I’m not really sure why… Anyhow this popped up in Andrew Wiles famous proof of Fermat’s Last Theorem. Fermat was great. He did lots of incredibly difficult and complicated maths in the margins of a textbook without ever showing any of his working out. It took mathematicians almost 400 years to figure it all out and I reckon he is probably the reason that maths teachers today insist that you show your working.
Back to the BSD conjecture (that’s its new hip name). It looks at equations that describe a particular type of graph. For a graph that is just a single straight line we have the general equation y = mx + c, where m is the gradient of the line and c is the intercept with the y-axis. Fancier graphs called elliptic curves also have equations describing them they just happen to be a little more complicated. To get an idea of what an elliptic curve is, just take a pencil (or heaven forbid a pen – drawing graphs with a pen is a big no-no in maths for some reason) and draw a nice squiggly line that has no sharp corners and doesn’t cross over itself. That is your very own elliptic curve (or near enough). Here are some actual elliptic curves for reference.
The BSD conjecture (catchy isn’t it?) asks whether or not the equations for these elliptic curves have solutions (the points where they cross the axes) that are whole numbers or fractions. Taking whole numbers and fractions together gives you a group of numbers called rational numbers. Almost every number you can think of is a rational number: 1, 2, 3, ½, ¼, 100 etc. but they are actually less common that the other type of number – irrational numbers. Irrational numbers are everything else that isn’t a whole number or a fraction. So here we mean pi, e, the square root of 2, the golden ratio, there are a few very famous ones and literally an unlimited amount of others.
In maths if we have a fixed number of something we say it is finite: no matter how few or how many we have, we are able to put them in some kind of order and count them up. It might take us forever to count them, say if we wanted to count every living thing on earth, but the idea is that if we kept going and didn’t stop (or die) then eventually we would have counted everything alive on earth and arrived at a final number. If we can’t count everything then we say it is infinite. The most obvious example are numbers themselves, or even just whole positive numbers. If you started counting right now you could continue forever. There is no end point, you would just keep counting higher and higher and higher and higher. You could have started counting at the beginning of the universe and you would still be going today. Numbers go on forever and for that reason they are infinite.
We are almost there so let’s take stock. We have fun squiggly curly graphs called elliptic curves. These are described by equations called elliptic equations. These equations can sometimes have rational solutions (the points where they cross the axes), which are fractions or whole numbers. We can have a finite (number we can count) or infinite (goes on forever – we can always find one more) number of these solutions. The BSD conjecture asks whether or not there is a simple way to tell if an elliptic equation has a finite or infinite number of rational solutions. There you go – who’d have thought it was as easy as that? “Simples” as that silly little Russian meerkat would say…
I’ve written a series of articles on each of the 7 Millennium Problems which can be found here.
Navier-Stokes Equations
Millennium Problem number four is my favourite, hands down. I’m probably not supposed to be biased but when you have an equation tattooed on your body the rules change. The Navier-Stokes equations describe the flow of every fluid you can possibly think of: rivers, water from a tap, waves, wind, air flow around an aeroplane, ice in glaciers, ketchup, honey dripping off a spoon, blood in your body… I could go on forever.
The fact that these equations can do all of this is great – it shows that things in nature behave similarly, and we may actually understand some of it. But, there is a downside. To be able to describe such a variety of different fluids all at once, these equations are super-complex. I’m talking the plot of Inception complex. And just like no-one really understands Inception (no matter what they might tell you), mathematicians don’t understand all the little intricacies of the Navier-Stokes equations.
The easiest way to think about it is using what we call a singularity. It might sound complicated but I’m going to explain it step-by-step so stick with me. Start with a number, let’s say 2. Divide 1 by 2 and you get 1/2. Now take a smaller number, say 1/4 and divide 1 by it – ‘dividing by a fraction is the same as multiplying by it upside down’ (sorry, I’m just hearing the voice of my primary school teacher). The answer is 4 though. Now take a smaller number, say 0.1 and divide 1 by it. You get 10. Take a smaller number 0.01 and divide 1 by it, you get 100. Continue this: divide 1 by smaller and smaller and smaller numbers and you will get a bigger and bigger and bigger answer. So what happens when you divide 1 by 0? Maths breaks is the answer, but we can think of it as infinity or in our case a singularity.
Singularities occur in nature too with perhaps the most famous example being a black hole. These guys are so complicated that even Stephen Hawking struggles to understand what’s going on, which gives you an idea of why singularities are such a nuisance. Going back to the Navier-Stokes equations and the motion of fluids, my favourite example involves bubbles. Let’s do a little experiment. Take two circular pieces of wire and holding them close together dip them in soapy water. Imagine those little bottles of bubbles you used to get as a kid, and the plunger thing with the circle bit on the end… that. Well, two of them close together. The idea is that when you hold them close together, dip them in soap and then take them out a bubble will form between the two. It should form a cylinder shape – like the centre of a toilet roll. As you move the two circular wires apart the bubble will stretch and grow taller (think toilet roll to kitchen roll). You can keep moving the wires further and further apart and the bubble gets longer and longer and then POP! You’ve moved them too far apart and the bubble breaks.
Thinking about this mathematically is a nightmare. It makes sense at first, the wires move further apart and the size of the bubble grows. Increase the distance between the wires and the bubble size increases – a nice simple mathematical relationship. Until you reach the point where the bubble pops. At this instant the increase in the distance between the wires causes a sudden and incredibly fast decrease in the bubble size to zero. It’s so fast you can call it infinite. This is your singularity. The video below shows a great example of the experiment I’ve just described and shows the moment where the bubble size suddenly goes to zero.
As I said above the Navier-Stokes equations model the flow of any and every fluid – this means they describe the bubble popping madness we’ve just looked at and most importantly the singularity. We don’t know how or why or what is going on with these guys – again, think of black holes – and that is the Millennium Problem. Can we improve our understanding of these equations? In Lord of the Rings it was one ring to rule them all, in the maths of fluids the Navier-Stokes equations are your ruler… now bow down and make some bubbles.
You can listen to me interviewing Professor Keith Moffat about the problem here.
I’ve written a series of articles on each of the 7 Millennium Problems which can be found here.
